Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:
import java.util.Scanner;

public class Main {

	static Scanner sc = new Scanner(System.in);
	static final long MOD = 1_000_000_007;

	public static void main(String[] args) {
		int N = sc.nextInt();
		double[] P = new double[N];
		for (int i = 0; i < N; i++) {
			P[i] = sc.nextDouble();
		}
		double[][] dp = new double[N + 1][N + 1];
		dp[0][0] = 1;
		for (int i = 0; i < N; i++) {
			for (int j = 0; j <= i; j++) {
				dp[i + 1][j + 1] += dp[i][j] * P[i];
				dp[i + 1][j] += dp[i][j] * (1 - P[i]);
			}
		}
		double ans = 0;
		for (int i = N; i > N / 2; i--) {
			ans += dp[N][i];
		}
		System.out.printf("%.10f\n", ans);
	}
}